{
  "id": "0807.2195",
  "version": "v1",
  "published": "2008-07-14T16:32:12.000Z",
  "updated": "2008-07-14T16:32:12.000Z",
  "title": "On the vanishing of the Rokhlin invariant",
  "authors": [
    "Tetsuhiro Moriyama"
  ],
  "comment": "27 pages, 1 figure",
  "categories": [
    "math.GT"
  ],
  "abstract": "It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant of an amphichiral integral homology 3-sphere M vanishes. In this paper, we give a new direct proof of this vanishing property. For such an M, we construct a manifold pair (Y,Q) of dimensions 6 and 3 equipped with some additional structure (6-dimensional spin e-manifold), such that Q = M \\cup M \\cup (-M) and (Y,Q) \\cong (-Y,-Q). We prove that (Y,Q) bounds a 7-dimensional spin e-manifold (Z,X) by studying the cobordism group of 6-dimensional spin e-manifolds and the Z/2-actions on the two--point configuration space of M minus one point. For any such (Z,X), the signature of X vanishes, and this implies the vanishing of the Rokhlin invariant. The idea of the construction of (Y,Q) comes from the definition of the Kontsevich-Kuperberg-Thurston invariant for rational homology 3-spheres.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-14T16:32:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57N70",
      "57R20",
      "55R80"
    ],
    "keywords": [
      "rokhlin invariant",
      "spin e-manifold",
      "two-point configuration space",
      "amphichiral integral homology",
      "casson invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.2195M"
    }
  }
}